Statistical Analysis

The data were analyzed using a bayesian logistic regression using the bernoulli family and logit linking function in the R package brms. The model priors followed the default in brms, using the Student’s T distribution with 3 degrees of freedom. A single model was fit to the data where the outcome variable was the probability of realizing the a stop as an approximant. The outcome was predicted by the population (fixed) effects of group (3 levels: labeled “control”, “experimental”, and “comparison” in the data), and session (3 levels: labeled “1”, “5” and “6”). The random effects included a random slope by participant for session. The model was run using with 2000 iterations of Hamiltonian Monte-Carlo sampling (1000 warm up), across 4 chains and 8 processing cores.

Results

Figure 1 shows a by group analysis of the raw data for the “control”, “experimental”, and “comparison” groups. The color of the points represent the category of each stop realization, while the y-axis shows the duration of that realization. The x-axis shows the spectral pattern of each stop (continuous (C), Stripe/Burst (S) or whitening (W)).

Figure 2 shows just the number of each type of realization per group per session. The figure shows that the number of approximant realizations are relatively low by all groups in session 1, while the quantity of approximant realizations was larger in both sessions 5 and 6 for the experimental groups, but not for the comparison and control groups.

Table 1 shows the output of the Bayesian logistic regression model. Overall, the model’s conditional r squared value suggests that the model explained 0.29 percent of the variance observed in the data. Figure 3 visualizes the model in a forest plot. Each density plot is a summary of the most probable parameter estimates from the Bayesian model, and the values on the right side of the plot are the 95% highest density interval in log-odds.

Table 1

Estimate Est.Error Q2.5 Q97.5
Intercept -3.5346760 0.8330690 -5.3776991 -2.122251
session5 0.8238395 0.8224129 -0.7753443 2.445888
session6 -0.6385882 1.0724133 -3.1063402 1.204214
groupControl 0.7301176 1.0885412 -1.3381893 2.962385
groupExperimental 0.8172550 1.1415364 -1.2994903 3.155540
session5:groupControl -1.0700826 1.1012484 -3.2886012 1.056761
session6:groupControl -0.1387219 1.3137497 -2.6082699 2.576936
session5:groupExperimental 0.9359264 1.0866427 -1.1764587 3.124677
session6:groupExperimental 2.6551138 1.2786853 0.4448966 5.514512

The output of logistic regressions models can be interpreted by adjusting the parameter estimates to a baseline group represented by the intercept of the model. In this case, the intercept of the model tells us the log-odds of an approximant realization by the comparison group at session 1. To figure out how the other groups perform, we simply add appropriate parameter estimates of the model to the intercept as it is appropriate. For example, by adding session 5 (.47) to our intercept (-3.19), we adjust for session while maintaining group (comparison). As a result, our estimate is a log odds of -2.72 of producing a token as an approximant at session 5 for the comparison group.

Rather than make arguments using log-odds, it’s also possible to convert this metric to probability. I’ve done so using the conditional_effects function from the brms package in R. The output of this function carries out the appropriate calculations and converts them to probability. This information is visualized in Figure 4 and the numerical output can be seen in Table 2. As can be seen in both, the experimental group saw an increase in the probability of approximant realization from both session 1 to session 5 and from session 5 to session 6. This increase does not occur in the comparison or control groups.

Table 2: Probability of an approximant realization at each session
group Session.1 Session.5 Session.6
Comparison 0.041 [0.009-0.128] 0.064 [0.014-0.208] 0.019 [0.001-0.084]
Control 0.06 [0.013-0.194] 0.047 [0.008-0.183] 0.03 [0.003-0.134]
Experimental 0.068 [0.012-0.229] 0.282 [0.069-0.626] 0.34 [0.094-0.692]

Individual analysis

In addition to the group analysis, an individual analysis of the probability of approximant realization at each time point was carried out using the random slope by participant for session in the Bayesian Model. First, Figure 5 shows duration, category and spectral information for each participant at each session for the control group. The same information can be seen for the comparison group in Figure 6 and the experimental group in Figure 7.

Figure 8 utilizes the random slope for session by participant from the Bayesian model to determine the probability of an approximant realization at each session by the experimental group. The figure shows that an approximant realization was more probable at session 5 than session 1 and at session 6 than session 5 for all participants except for participant 1 and participant 31. For participant 31, although the probability of an approximant realization was lower at session 6 0.539 [95% HDI 0.049 - 0.968] than session 5, (0.72 [95% HDI 0.097 - 0.995]), both of these probabilities were higher than session 1 (0.126 [95% HDI 0.024 - 0.469]). Table 3 lists all probabilities of approximant realizations by each participant at each timepoint, with the upper and lower bounds of the Highest Density Interval.

Table 3a: Experimental Group - All probabilities of approximant realizations by each participant at each session.

participant session Probability Lower Upper
1 Session 1 0.096 0.021 0.350
8 Session 1 0.064 0.013 0.252
11 Session 1 0.288 0.078 0.721
12 Session 1 0.032 0.005 0.148
13 Session 1 0.044 0.009 0.201
28 Session 1 0.008 0.001 0.049
31 Session 1 0.126 0.024 0.469
1 Session 5 0.328 0.017 0.910
8 Session 5 0.304 0.018 0.926
11 Session 5 0.735 0.100 0.990
12 Session 5 0.116 0.003 0.755
13 Session 5 0.143 0.004 0.783
28 Session 5 0.026 0.000 0.550
31 Session 5 0.720 0.097 0.995
1 Session 6 0.424 0.041 0.919
8 Session 6 0.362 0.036 0.918
11 Session 6 0.798 0.199 0.993
12 Session 6 0.209 0.013 0.859
13 Session 6 0.244 0.015 0.852
28 Session 6 0.038 0.000 0.531
31 Session 6 0.539 0.049 0.968

Table 3b: Experimental Group - All probabilities of approximant realizations by each participant at each session.

participant session Probability Lower Upper
6 Session 1 0.122 0.027 0.462
17 Session 1 0.018 0.001 0.124
32 Session 1 0.075 0.013 0.336
42 Session 1 0.017 0.001 0.114
43 Session 1 0.120 0.023 0.427
44 Session 1 0.071 0.012 0.314
50 Session 1 0.045 0.006 0.256
51 Session 1 0.096 0.019 0.390
6 Session 5 0.060 0.001 0.652
17 Session 5 0.011 0.000 0.402
32 Session 5 0.068 0.002 0.763
42 Session 5 0.011 0.000 0.373
43 Session 5 0.176 0.008 0.918
44 Session 5 0.062 0.002 0.734
50 Session 5 0.024 0.000 0.482
51 Session 5 0.113 0.005 0.866
6 Session 6 0.093 0.007 0.880
17 Session 6 0.007 0.000 0.202
32 Session 6 0.049 0.003 0.712
42 Session 6 0.007 0.000 0.176
43 Session 6 0.055 0.002 0.582
44 Session 6 0.032 0.001 0.438
50 Session 6 0.019 0.000 0.347
51 Session 6 0.042 0.001 0.485

Table 3c: Comparison Group - All probabilities of approximant realizations by each participant at each session.

participant session Probability Lower Upper
3 Session 1 0.021 0.002 0.120
7 Session 1 0.046 0.006 0.232
14 Session 1 0.010 0.001 0.073
16 Session 1 0.049 0.009 0.276
18 Session 1 0.010 0.001 0.073
35 Session 1 0.010 0.001 0.076
39 Session 1 0.042 0.007 0.234
41 Session 1 0.351 0.090 0.802
47 Session 1 0.022 0.003 0.140
49 Session 1 0.021 0.003 0.124
3 Session 5 0.054 0.001 0.717
7 Session 5 0.187 0.008 0.932
14 Session 5 0.016 0.000 0.460
16 Session 5 0.058 0.001 0.664
18 Session 5 0.016 0.000 0.458
35 Session 5 0.016 0.000 0.454
39 Session 5 0.083 0.002 0.765
41 Session 5 0.631 0.048 0.990
47 Session 5 0.056 0.001 0.748
49 Session 5 0.054 0.001 0.731
3 Session 6 0.011 0.000 0.231
7 Session 6 0.026 0.001 0.438
14 Session 6 0.005 0.000 0.157
16 Session 6 0.037 0.002 0.755
18 Session 6 0.005 0.000 0.155
35 Session 6 0.005 0.000 0.157
39 Session 6 0.022 0.001 0.356
41 Session 6 0.335 0.020 0.983
47 Session 6 0.012 0.000 0.248
49 Session 6 0.011 0.000 0.238

Appendix/extras

Here I’ve included the plots of the three groups removed from the main analysis, Comparison-High”, “Control-High-Partial”, and “Experimental-High”. From these visualizations, it can be seen that the groups are overall producing approximants at a high rate in the beginning (session 1), which seems like good rationale for their removal.

Finally, the session 2 analyses are reported here. Figure 11 shows the total percentage of approximant realizations by 4 groups at session 2. Table 4 gives this information in text.

Table 4: Amount and percentage of each realization type during all sessions

category group session n mean_ri sd_ri mean_wmrv sd_wmrv mean_duration sd_duration total percent
approximant Comparison 1 6 8.19 8.04 6.02 6.73 23.29 9.61 76 0.08
stop Comparison 1 5 26.19 6.94 10.99 1.60 42.19 9.68 76 0.07
tap Comparison 1 65 10.51 6.44 6.79 3.89 17.99 3.59 76 0.86
approximant Comparison-High 1 12 12.71 6.00 6.51 2.19 33.43 8.40 15 0.80
tap Comparison-High 1 3 8.56 4.31 4.77 1.64 24.16 3.91 15 0.20
approximant Control 1 5 5.57 3.89 4.10 3.58 24.30 9.00 59 0.08
stop Control 1 5 18.74 7.03 18.03 12.18 44.69 15.43 59 0.08
tap Control 1 49 12.56 6.39 9.34 4.97 17.84 3.56 59 0.83
approximant Control-High-Partial 1 6 4.77 3.51 2.48 1.53 38.95 12.01 6 1.00
approximant Experimental 1 5 11.90 8.09 6.93 5.76 26.31 9.27 52 0.10
tap Experimental 1 47 11.78 6.41 8.16 3.97 17.51 3.77 52 0.90
approximant Experimental-High 1 25 10.27 7.02 5.83 3.91 32.04 6.39 31 0.81
stop Experimental-High 1 2 12.20 5.81 7.93 4.98 35.50 1.37 31 0.06
tap Experimental-High 1 4 14.67 3.11 7.42 2.04 20.99 3.68 31 0.13
approximant Comparison 2 33 18.39 7.52 11.77 5.49 50.73 15.58 50 0.66
stop Comparison 2 9 25.08 7.79 17.06 5.89 58.10 10.48 50 0.18
tap Comparison 2 8 13.41 5.95 9.69 3.72 22.77 6.13 50 0.16
approximant Comparison-High 2 10 13.30 4.32 8.01 3.43 56.66 15.00 10 1.00
approximant Experimental 2 17 15.58 10.93 8.91 6.85 52.79 12.95 30 0.57
stop Experimental 2 9 31.12 5.97 23.20 5.61 77.09 11.58 30 0.30
tap Experimental 2 4 15.40 5.26 12.77 0.93 20.35 5.29 30 0.13
approximant Experimental-High 2 16 19.02 6.74 8.78 4.91 59.54 15.23 20 0.80
stop Experimental-High 2 4 21.76 7.44 20.53 9.49 50.76 8.19 20 0.20
approximant Comparison 5 14 13.39 10.69 7.63 6.42 41.52 11.60 108 0.13
stop Comparison 5 27 21.48 7.13 19.22 8.09 41.58 7.52 108 0.25
tap Comparison 5 67 13.71 6.24 10.29 4.93 18.03 4.49 108 0.62
approximant Comparison-High 5 20 11.15 5.47 7.65 5.55 37.88 9.06 22 0.91
stop Comparison-High 5 1 9.63 NA 4.61 NA 48.41 NA 22 0.05
tap Comparison-High 5 1 0.15 NA 1.71 NA 28.68 NA 22 0.05
approximant Control 5 7 8.16 6.15 4.20 2.20 28.43 11.09 84 0.08
stop Control 5 5 20.46 5.20 11.75 2.83 40.96 5.72 84 0.06
tap Control 5 72 8.99 5.44 5.21 2.74 16.25 4.84 84 0.86
approximant Experimental 5 24 11.81 9.09 8.07 7.15 37.65 11.21 70 0.34
stop Experimental 5 11 26.15 5.86 20.77 6.18 50.44 12.39 70 0.16
tap Experimental 5 35 12.76 7.53 9.56 4.94 18.25 4.12 70 0.50
approximant Experimental-High 5 28 9.40 6.91 5.72 4.54 32.37 9.11 31 0.90
tap Experimental-High 5 3 13.72 2.48 9.45 1.62 20.59 7.95 31 0.10
approximant Comparison 6 4 7.76 2.84 3.22 0.50 31.21 12.53 77 0.05
stop Comparison 6 19 20.02 7.86 10.96 5.54 41.34 9.39 77 0.25
tap Comparison 6 54 12.12 5.24 6.48 2.16 19.07 4.23 77 0.70
approximant Comparison-High 6 11 12.93 4.62 4.57 1.45 32.96 10.97 15 0.73
stop Comparison-High 6 1 20.96 NA 12.02 NA 36.87 NA 15 0.07
tap Comparison-High 6 3 15.15 4.36 9.03 2.72 21.20 5.07 15 0.20
approximant Control 6 3 7.41 6.21 4.18 2.61 17.42 1.43 56 0.05
stop Control 6 2 25.01 6.58 12.78 7.21 40.45 9.32 56 0.04
tap Control 6 51 11.22 5.53 5.76 2.55 17.09 3.99 56 0.91
approximant Control-High-Partial 6 3 8.51 4.55 3.67 2.18 26.60 7.98 8 0.38
tap Control-High-Partial 6 5 16.03 7.09 9.24 3.01 18.40 6.27 8 0.62
approximant Experimental 6 19 7.66 5.26 3.20 2.40 27.48 8.59 52 0.37
stop Experimental 6 8 20.12 5.24 11.03 2.43 44.84 9.94 52 0.15
tap Experimental 6 25 9.88 4.71 5.27 2.49 20.63 4.36 52 0.48
approximant Experimental-High 6 19 9.40 9.00 4.64 4.21 40.27 15.11 31 0.61
stop Experimental-High 6 4 25.32 7.16 13.07 2.45 62.91 6.21 31 0.13
tap Experimental-High 6 8 15.70 8.55 9.40 4.75 23.38 4.10 31 0.26

Conclusion and takeaways

Overall, it seems that there is evidence that the experimental group, but not the control or comparison groups, increased in the probability of realizing a given stop as an approximant as a function of session. Additionally, this group maintained their improvement from session 5 to session 6, although this difference was not as big as the initial change from session 1 to session 5. Individual analyses corroborate the trends seen at the group level, with 7 out of 8 members of the experimental group being more likely to produce an approximant after session 1. However, one must be cautious in the interpretation of these results due to the lower sample observed here. The HDIs of the parameter estimates suggest that there are rather wide amounts of uncertainty around the probability estimates. As a result, I recommend tempering strong narrative conclusions on the basis of this data.